Questions tagged [numerics]
The numerics tag has no usage guidance.
783
questions
0
votes
1answer
66 views
Numerical solution for gradient(slope)
Abstract
I have the next equation to find a force, for my problem:
$$U=-\int \vec{m}\small{(x)}\times \vec{B}(x)dV$$
$$\vec{F}=-\nabla U$$
Considering 3-dimensional space with x,y,z coordinates, ...
17
votes
4answers
3k views
Is half precision supported by modern architecture?
I am new to computer science and I was wondering whether half precision is supported by modern architecture in the same way as single or double precision is. I thought the 2008 revision of IEEE-754 ...
1
vote
0answers
46 views
Plot of ratio of two integrals:
Consider the following integrals
$$
I_1(x) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy) − F(x −\mathrm iy)}{\mathrm e^{2πy}-1},
$$
And
$$I_2(x) =\int_1^x F(t)dt$$
Where, $ F(z) = \sin^2[π\...
1
vote
3answers
111 views
Runge-Kutta method for an ODE with initial value which is root of denominator
I wrote a code in Fortran to solve this differential equation using RK4 method:
$$
\frac{dy}{dx}=A\sqrt{\frac{B}{y}+\frac{C}{y^2}}
$$
$A$, $B$, and $C$ are some known constants. The problem is that ...
5
votes
0answers
90 views
How to numerically evaluate this double Integral?
I want to evaluate the following integral:
$$\int_{0}^{60} \ \left(\int_{0}^{2z} 0.5\cdot t \left(\mathrm{erf}(t-a) -1 \right)J_{0}(qt)\mathrm{d}t \right)^2 \mathrm{exp}\left(-\frac{(z-a)^2}{2s^2}\...
2
votes
3answers
163 views
Comparison of integrals with a function:
Consider the following integral:
$$S(q)=\int_{x=2}^q\sin^2\left(\frac{π\Gamma(x)}{2x}\right)dx$$
And consider the functions :
$$R(q)=\frac{q}{\log(q)}$$
$$T(q)=\int_2^q\frac{1}{\log(x)}dx$$
I ...
0
votes
0answers
52 views
Difficulty to prove the coercivity of $a(u, v)=\int_{\Omega} \nabla u \cdot \nabla v$
I'm stuck at proving the coercivity of the bilinear functional in the variational formulation of the problem:
\begin{array}{c}
-\nabla^{2} u=f \quad \text { in } \Omega \\
u=g_{D} \text { on } \...
0
votes
0answers
27 views
Implementation of nonlinear optimization for Generalized Nash-Equilibrium
I have to find a solver for $\begin{equation}
\min_{x^{\nu}} \Theta_{\nu}(x^{\nu},x^{-\nu})
\end{equation}$ with $x^{\nu} \in X_{\nu}$ which is a convex set.
$x^{*}$ needs to satisfy $$\nabla_{x^{\nu}...
0
votes
0answers
21 views
Markov chain Monte Carlo with stopping time
I asked the same question two days ago on MSE, but received no answer. So I post it here in hope to get any suggestion. As long as I have answer, I will close the other one.
Let $(X_t)$ be a ...
1
vote
1answer
74 views
Why is my Cahn-Hilliard simulation separating out so finely?
I am trying to simulate the Cahn-Hilliard equation using Python, but the 2 fluids aren't separating into big blobs, as desired, under any conditions. I'm setting up (what I think is) an orthogonal ...
0
votes
0answers
15 views
Shallow water equations: boundary conditions for sub- and super-critical flow
This question is (sort of) a continuation of this previously asked question. I am wondering about, in general, how we construct well-posed boundary conditions (both continous and numerical) for flow ...
1
vote
0answers
36 views
Advice for a topic in a seminar
I am a master student in Mathematics, and I have to prepare a seminar for a course in mathematical methods for applied sciences. I have a good background in numerical analysis for ODEs, PDEs and hence ...
0
votes
3answers
69 views
Changing variables in integral to avoid infinity
I want to write a code in Fortran to solve this integral numerically: $$\int_{1095}^\infty \frac{dx}{x\sqrt{(x+644.153)(4.17 \cdot 10^{-5} x+0.145)}}$$ What is the best method for it?
I tried Monte-...
6
votes
1answer
243 views
What kind of a researcher am I?
So far, I've worked a bit in modeling, simulations and simple lab experiments, and I've really enjoyed all three research methods to approach a single research question. I can write tricky (in terms ...
1
vote
0answers
25 views
Unable to achieve semi-linear running time in computation of continuant
I am trying to compute the continuant of a list of numbers $a_0, a_1,...,a_n$, defined by the recursion relation: $K_{n+1} = a_{n+1} K_n + K_{n-1}$ and $K_0 = 1$ (see Wikipedia).
I am trying to use ...
1
vote
0answers
19 views
Convection equation expanded in Legendre polynomials
In some areas of physics, for example radiation transport in plasmas, it is beneficial to re-express the convection equation in terms of a Legendre polynomial expansion. This results in a set of ...
1
vote
1answer
53 views
Galerkin method for heat equation
I'm working out the Galerkin method for the heat equation $$\frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = 0$$ subject to $u(0,t)=0,u_x(1,t)=v(t)$.
I want to use a Fourier basis ...
0
votes
1answer
36 views
Coupled pdes of the first order
May question is about possible approaches to solve the following system
$$
\begin{array}{rcl}
\nabla{n}&=&n\,\mathbf{E},\\
\nabla\cdot\mathbf{E}&=&1-n,
\end{array}
$$
in general with ...
1
vote
0answers
52 views
Bifurcation points on homotopy path by numerical continuation?
I am trying to implement an algorithm that finds (possibly) all solutions of a system of nonlinear polynomial equations $$F(X) = 0$$ I thought about using the (convex) homotopy $$H(X,T) = TF(X) + (1-T)...
2
votes
0answers
69 views
Numerical method for harmonic oscillator with jumping constant
Let $k_1 \neq k_2$ be positive reals, $t_0 > 0$ and consider the following Cauchy problem in $[0,+\infty)$:
\begin{cases}
y(t) + k(t)y''(t) = 0 \newline
y(0) = 1/\sqrt{k_1} \newline
y'(0) = 0,
\end{...
2
votes
1answer
127 views
Solving numerically a linear ODE
I start by saying that I do not have a strong background in numerical analysis, so I may miss some basic things or make trivial mistakes.
Motivated by some problems in digital signal processing, I ...
3
votes
1answer
77 views
Can you compare integer part of two fractions without division?
Suppose we need to compare
$\left \lfloor{a / b}\right \rfloor $
and
$\left \lfloor{c / d}\right \rfloor $
.
One way would of course be to calculate $a/b$ and $c/d$ by division. Is their a faster way?
1
vote
0answers
27 views
Tackling multiscale problem in numerical simulation
In a dusty plasma system there are more than one component with different masses, i.e, electrons, ions,neutrals and dust grains. Accordingly, there are more than one temporal and spatial scales ...
0
votes
1answer
95 views
Time complexity of numerical finite differences
I have a function $f:\mathbb R^N\to \mathbb R$ and I would like to compute all the partial derivatives of $f$ w.r.t. the $N$ input. What is the computational complexity using the (ones-sided) finite ...
1
vote
2answers
51 views
What space-time points should a known coefficient function be evaluated at when using the Lax-Friedrichs scheme to solve the transport equation?
For a scalar quantity $u = u(x, t)$, I'm considering the transport equation
\begin{align}
u_t + au_x &= 0, \qquad{x\in[0, L], \ t\in[0, T]}
\\
u(0, t) &= u_{\text{in}},
\\
u(x, 0) &= f(x).
...
0
votes
1answer
92 views
Normalized legendre and quadrature basis for discontinous Galerkin method
I've successfully implemented a 1D DG code with non-normalized Legendre basis and I've now moved onto developing a 2D code using tensor products. For my 2D code I've chosen to have normalized ...
0
votes
1answer
45 views
Reinch's modification to the Clenshaw recurrence gives no improvement
The classical Clenshaw recurrence (see Algorithm 3.1 here) is less accurate as $x\to \pm 1$. So Reich proposed a modification to it, which is discussed as Algorithm 3.2 as well as by Oliver.
While ...
6
votes
2answers
210 views
Numerical flux and source term in FVM (Burger's like equation)
I'm trying to solve the following equation with FVM
$$u_t + f(u)_x = g(u)$$
where $g$ is some smooth function of $u$ and $f(u) = \frac{u^2}{2}$. This is really similar to Burger's equation, except ...
1
vote
0answers
62 views
Compute mass matrix in vibrations problem by using finite element method
I have to compute the mass matrix of a Hexahedral mesh.
There are 3 methods to compute mass matrix. I'm interested in one method which consists of dividing the mass of the element by the number of ...
0
votes
1answer
40 views
Showing stability of numerical scheme: Decaying norm implies stability?
I have a little trouble formulating my question since I am not really sure what conclusions I am supposed to draw from the results I have obtained. I am sorry for the long problem formulation below, ...
4
votes
1answer
121 views
Which optimization method can be used to do the following?
I've the following system of equations for studying information flow in the below graph,
$$ \frac{d \phi}{dt} = -M^TDM\phi + \text{noise effects} \hspace{1cm} (1)$$
Here, M is the incidence ...
0
votes
1answer
66 views
Minimizing vector norm without using symbolic packages
I am new to octave and still learning what I can and cannot do with it.
I am asked to write a following function ('ltrigp'):
[a, b, c, info]=ltrigp(x,y)
Where x, ...
3
votes
2answers
139 views
Mass conservation for hyperbolic relaxation problem
I have solved numerically the following system:
\begin{cases}
\partial_t{u} + \partial_x{v} = 0 \\
\partial_t{v} + \frac{1}{\varepsilon^2}\partial_x{u} = -\frac{1}{\varepsilon^2}(v-f(u))
\end{cases}
...
2
votes
1answer
54 views
Using linear regression to find the ideal point given a set of trajectory's data
I have a set of points in 2D obtained from a pendular movement with some noise. I want to determine where is equilibrium point ($x_0$, $y_0$) from which the rope is fixed. There are at least two ...
2
votes
1answer
120 views
Gradient-jump penalty term in FEM
I am slightly confused regarding the meaning of the $i-th$ gradient-jump term $[\nabla \phi_i]$ in the context of finite element methods, used in the assembly of the stiffness matrix (an example with <...
2
votes
4answers
131 views
Testing the time dependent Schrodinger Equation with an analytical solution?
I am numerically solving the Schrodinger Equation in 1D first and in higher dimension later,
but I want to know the convergence rate of my numerical solver in different grid size and numerical methods....
2
votes
0answers
40 views
Evaluating integral $F(r_2) = \frac{1}{r_2^n} \int_0^{r_2} r_1^n f(r_1)\ \mathrm{d}r_1$ without growing instability
I have the following expression to be numerically integrated in a vector-based library (e.g. numpy, MATLAB, etc),
$$
F(r_2) = \frac{1}{r_2^n} \int_0^{r_2} r_1^n f(r_1)\ \mathrm{d}r_1,
$$
where $n$ is ...
4
votes
1answer
174 views
The velocity Verlet method and variable time steps
Does the velocity Verlet handle variable time steps? I found controversial statements about it.
In the paper Skeel, R. D., "Variable Step Size Destabilizes the Stömer/Leapfrog/Verlet Method", BIT ...
1
vote
0answers
61 views
Are linear, CTCS codes always stable?
I would like to solve some equations which basically look like this
$$\frac{\partial u}{\partial x}=F\left(v,\frac{\partial v}{\partial y},\frac{\partial^2 v}{\partial y^2}\right),$$
$$\frac{\partial ...
1
vote
0answers
35 views
Comparison of convection time - theoretical value vs computed
This is a follow up to my previous post here,
I'm solving for convection in 1D
$$\frac{\partial C}{\partial t} = - v\frac{\partial C}{\partial x}$$
The discretization of the above equation is ...
1
vote
0answers
44 views
Unsteady diffusion equation with spatial finite elements and Forward Euler in time
I have solved the unsteady diffusion equation using piece-wise linear Finite elements(triangles) for spatial discretisation and Forward Euler for temporal discretisation. I have the following mesh ...
4
votes
1answer
93 views
Parareal for particle simulations
Recently I have stumbled upon this video of M. J. Gander https://www.youtube.com/watch?v=dn5vqN8ezuE
and the coresponding notes that he wrote on Time Parallel Time Integration and I find it a quite ...
0
votes
1answer
92 views
Question on comparing the accuracy of numerical schemes
This is a follow up to my previous post here
I'm solving the following 1D transport equation .
$$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2}-v\frac{\partial C}{\partial x}$$...
2
votes
1answer
129 views
Question on how MATLAB's pdepe solver works
I'm solving the following 1D transport equation in MATLAB's pdepe solver.
$$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2}-v\frac{\partial C}{\partial x}$$
At the inlet (left ...
4
votes
2answers
125 views
Effect of mesh size on solution curves for a 1D problem
I'm interested in studying the effect of mesh size on the behavior of the solution curves of 1D convection-diffusion problem.
$$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2} - ...
2
votes
1answer
104 views
Slope limiting for discontinuous Galerkin (DG) method
I had a question regarding the implementation of the TVB limiter for the RKDG method by Cockburn. I have seen that some implementations of the DG method use normalized Legendre polynomials such that ...
1
vote
0answers
59 views
Conjugacy in Non-linear Conjugate Gradient Descent
In linear conjugate gradient method, our goal is to solve the system of linear equations $$Ax = b$$ where A is a symmetric positive definite matrix, and that is equivalent to finding the minimizer of ...
1
vote
1answer
33 views
How to compute the mean 2D slice of a 3D set of data in MPI
I have a 3D set of data v(i,j,k), and I want to compute the mean 2D slice vmean(i,j) summing up the ...
1
vote
0answers
54 views
How to minimize a integral function using a constant step gradient method in Python?
I am developing a practical work of the following system of ode
\begin{align}x'(t) &= k_1h(t) - (k_2+k_3)x(t)\\
y'(t) &= k_3x(t)\end{align}
and $z(t) = (1-k_4)(x(t)+y(t))+k_4h(t)$, where $h(...
3
votes
1answer
188 views
log-sum-exp trick for signed/complex numbers
I need to evaluate a sum of values that are on many different orders of magnitude in scale but might be signed. I’ve had great luck with the “log-sum-exp” trick for an unsigned version of my problem, ...
